How Much of a Rubik's Cube Does a Three-Face Photograph Determine?

Corrected and Expanded Version 2 — 31 October 2026 This paper investigates how much information is required to reconstruct the complete state of a standard 3×3×3 Rubik's Cube from photographs or observations of its faces. It is a substantially revised, corrected, and extended version of the original publication (Eigler, 2026, DOI: 10.5281/zenodo.23831559). Correction of the original publication The original version incorrectly claimed in Theorem 13.1 and Corollary 13.2 that five completely observed faces always uniquely determine the sixth face and, consequently, that two appropriately registered photographs revealing at least five faces guarantee complete reconstruction. These claims are explicitly withdrawn. The present version provides a concrete counterexample consisting of two distinct, legally reachable Rubik's Cube configurations that agree on five complete faces but differ on the sixth. The counterexample establishes that five observed faces do not guarantee unique reconstruction. New mathematical results The corrected and expanded analysis establishes the following: Three observed faces: Three mutually adjacent, completely monochromatic faces with their correct centre colours admit exactly 96 legally reachable completions. Four observed faces: A new deductive proof establishes that no observation of four complete faces can uniquely determine a legally reachable cube state. The proof covers both geometrically distinct cases: two adjacent hidden faces and two opposite hidden faces. Five observed faces: Unique reconstruction is not guaranteed, as demonstrated by an explicit pair of distinct reachable configurations with identical observations on five faces. Six observed faces: Complete observation guarantees unique reconstruction. Consequently, six complete faces are necessary and sufficient for a universal guarantee of unique reconstruction, although fewer faces may suffice for particular configurations. Additional mathematical and computational content The paper retains and integrates the original results on universal numerical state indexing, feasibility classification, exact completion counting, rotational symmetry, Burnside's lemma, and the distinction between physically assemblable and legally reachable configurations. The numerical encoding is now explained directly in the main text through a numbered cube net, a fixed colour-code convention, and a unique base-six representation of the 48 noncentral facelets. Complete reference programs, computational checks, and reproducibility materials are provided in the accompanying source archive. Publication history and transparency This version supersedes the mathematical reconstruction claims of the original publication wherever they conflict with the corrected results. The original version remains identifiable through its DOI, preserving a transparent record of the correction. The correction does not withdraw the original 96-completion theorem or the unaffected state-indexing and counting results.